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I think that for a time-independent Hamiltonian $\:\hat{\mr H}\plr{t}\e\hat{\mr H}\:$ we could derive the exponential expression of the time evolution operator
\begin{equation}
\mr U\plr{t,t_0}\e e^{\m i\,\hat{\mr H}\plr{t\m t_0}/\hbar}
\tl{01}
\end{equation}
using its property
\begin{equation}
\mr U\plr{t,t_0}\e \mr U\plr{t,t_k}\mr U\plr{t_k,t_0}
\tl{02}
\end{equation}
and the identity
\begin{equation}
\lim_{n\bl\rightarrow\infty}\plr{1\p\dfrac{z}{n}\,}^n \e e^z\qquad \plr{n\bl\in \mathbb N\,, z\bl\in \mathbb C}
\tl{03}
\end{equation}
So, suppose that we insert in the time interval $\:\plr{t_0,t}\:$ a number of $\:\plr{n\m 1}\:$ time moments $\: \plr{n\bl\in\mathbb N}\:$ dividing it into $\:n\:$ sub-intervals
\begin{equation}
t_0\les t_1\les t_2\les\cdots\les t_k\les\cdots\les t_n\bl\equiv t
\tl{04}
\end{equation}
Using the property \eqref{02} we can build the operator $\:\mr U\plr{t,t_0}\:$ from a product of $\:n\:$ operators $\:\mr U\plr{t_k,t_{k\m 1}}\:$
\begin{align}
\mr U\plr{t,t_0}&\e\mr U\plr{t_n,t_0} \e\mr U\plr{t_n,t_{n\m 1}}\mr U\plr{t_{n\m 1},t_0}\e\mr U\plr{t_n,t_{n\m 1}}\mr U\plr{t_{n\m 1},t_{n\m 2}}\mr U\plr{t_{n\m 2},t_0}
\nonumber\\
&\e\mr U\plr{t_n,t_{n\m 1}}\mr U\plr{t_{n\m 1},t_{n\m 2}}\cdots\mr U\plr{t_k,t_{k\m 1}}\cdots\mr U\plr{t_3,t_2}\mr U\plr{t_2,t_1}\mr U\plr{t_1,t_0}
\tl{05}
\end{align}
that is
\begin{equation}
\mr U\plr{t,t_0}\e \prod\limits_{k\e 1}^{k\e n}\mr U\plr{t_k,t_{k\m 1}}
\tl{06}
\end{equation}
Now, make the $\:\plr{n\m 1}\:$ time moments equidistant
\begin{align}
t_k\m t_{k\m 1}&\e \Delta t \vp
\tl{07a}\\
\Delta t&\e\dfrac{t\m t_0}{n}
\tl{07b}
\end{align}
Taking $\:n\:$ extremely very large the time interval $\:\Delta t\:$ is infinitesimally very small so we could write
\begin{align}
t_k\m t_{k\m 1}&\e \mr dt \vp
\tl{08a}\\
\mr dt&\e\dfrac{t\m t_0}{n}\qquad \plr{n\gr\gr 1}
\tl{08b}
\end{align}
and equation \eqref{06} yields
\begin{equation}
\mr U\plr{t,t_0}\e \prod\limits_{k\e 1}^{k\e n}\mr U\plr{t_{k\m 1}\p \mr dt ,t_{k\m 1}}
\tl{09}
\end{equation}
But every factor in the rhs of \eqref{09} is an infinitesimal time evolution operator of the form $\:\mr U\plr{t\p \mr dt ,t}\:$ which satisfies the following equation
\begin{equation}
\mr U\plr{t\p \mr dt ,t}\bl\approx 1\!\!1\m\dfrac{i}{\hbar}\hat{\mr H}\,\mr dt
\tl{10}
\end{equation}
as proved and explained in the APPENDIX. Note that $\:1\!\!1\:$ is the identity operator.
So, we have
\begin{equation}
\mr U\plr{t_{k\m 1}\p \mr dt ,t_{k\m 1}}\bl\approx 1\!\!1\m\dfrac{i}{\hbar}\hat{\mr H}\,\mr dt\e 1\!\!1\m\dfrac{i\,\hat{\mr H}\plr{t\m t_0}/\hbar}{n}\qquad \plr{k\e1,2,\cdots,n}
\tl{11}
\end{equation}
and from \eqref{09} we obtain
\begin{equation}
\mr U\plr{t,t_0}\bl\approx \blr{1\!\!1\m\dfrac{i\,\hat{\mr H}\plr{t\m t_0}/\hbar}{n}}^n
\tl{12}
\end{equation}
Using the identity \eqref{03} we prove the exponential expression \eqref{01} of the time evolution operator
\begin{equation}
\mr U\plr{t,t_0}\e \lim_{n\bl\rightarrow\infty}\blr{1\!\!1\m\dfrac{i\,\hat{\mr H}\plr{t\m t_0}/\hbar}{n}}^n \e e^{\m i\,\hat{\mr H}\plr{t\m t_0}/\hbar}
\tl{13}
\end{equation}
$\hebl$
APPENDIX : $\:\texttt{The infinitesimal time evolution operator }\mr U\plr{t\p \mr dt ,t}$
The Schrodinger equation
\begin{equation}
i\,\hbar\dfrac{\mr d\vra{\psi\plr{t}}}{\mr dt}\e \hat{\mr H}\plr{t}\vra{\psi\plr{t}}
\tl{A-01}
\end{equation}
is linear in $\:\vra{\psi\plr{t}}\:$ so we expect this state to be obtained from an initial state $\:\vra{\psi\plr{t_0}}\:$ at time $\:t_0\:$ via a linear operator
\begin{equation}
\vra{\psi\plr{t}}\e \mr U\plr{t,t_0}\vra{\psi\plr{t_0}}
\tl{A-02}
\end{equation}
the time evolution operator $\:\mr U\plr{t,t_0}$. Since \eqref{A-02} would be valid equally well for the evolution of $\:\vra{\psi\plr{t}}\:$ from time $\:t\:$ to time $\:t\p\mr dt\:$ we have
\begin{equation}
\boxed{\:\:\vra{\psi\plr{t\p\mr dt}}\e \mr U\plr{t\p\mr dt,t}\vra{\psi\plr{t}}\:\:\Vp{\tfrac{\dfrac{a}{b}}{\dfrac{a}{b}}}}
\tl{A-03}
\end{equation}
The operator $\:\mr U\plr{t\p\mr dt,t}\:$ is the infinitesimal time evolution operator.
From equation \eqref{A-01} we have
\begin{equation}
\!\!\!\!\!\!\!\!i\hbar\dfrac{\mr d\vra{\psi\plr{t}}}{\mr dt}\e \hat{\mr H}\plr{t}\vra{\psi\plr{t}}\bl\implies\vra{\psi\plr{t\p\mr dt}}\m\vra{\psi\plr{t}}\bl\approx \m \dfrac{i}{\hbar}\,\hat{\mr H}\plr{t}\vra{\psi\plr{t}}\mr dt
\tl{A-04}
\end{equation}
so
\begin{equation}
\boxed{\:\:\vra{\psi\plr{t\p\mr dt}}\bl\approx \blr{1\!\!1\m \dfrac{i}{\hbar}\,\hat{\mr H}\plr{t}\mr dt}\vra{\psi\plr{t}}\:\:\Vp{\tfrac{\dfrac{a}{b}}{\dfrac{a}{b}}}}
\tl{A-05}
\end{equation}
where $\:1\!\!1\:$ is the identity operator.
Comparing equations \eqref{A-03},\eqref{A-05} we obtain
\begin{equation}
\boxed{\:\:\mr U\plr{t\p\mr dt,t}\bl\approx 1\!\!1\m \dfrac{i}{\hbar}\,\hat{\mr H}\plr{t}\mr dt\:\:\Vp{\tfrac{\dfrac{a}{b}}{\dfrac{a}{b}}}}
\tl{A-06}
\end{equation}
We note that because of \eqref{A-06} equation \eqref{10} is valid for time-dependent Hamiltonian in general.
